


L r^ . 






m-^m 



'^. . ^'^' "^H 



4>^^^ ^' :?XF w;- ■•; •■ 







'/■.jt^ 






^^q) 






LIBRARY OF CONGRESS, 



&lstitZ~... @ojtijri5^1 !f 0.. 



Shelf.f.^.i'^ 





'^m^ 



I 



THE PRACTICE 



OF 



MECHANICAL DRAWING 



FOR 



SELF-INSTRUCTION, 



BY 

y 

WILLIAMS WELCH, 

Instructor of Drawing at Clemson College. 



NEWBERRY, S. C. 

Elbert H. Aull, Puewsher and Printer, 

1895. 






<<.^ 



Copyright, 1895, by Wms. Welch. 
All Rights Reserved. 



PREFACE. 



This book is intended to enable mechanics and others to learn to make me- 
chanical drawings when thej^ cannot have the assistance of a teacher; and it is 
also intended as a text-book for beginners in mechanical drawing. 

Geometrical terms and problems, which are not needed in ordinary work,, 
have been avoided. 

WMS. WEI.CH. 
C1.EMS0N C0L1.EGE, s. c, 
Febrtiary 21st, 18 g^. 



CHAPTER IL 

CONSTKUCTION OF GEOMETEICAL PEOBLEMS. 

A draftsman should be familiar with the problems in this chapter, and should 
be able to construct them very accurately when necessary. In practice, they are 
sdldom used: but unless he knows how to draw them with precision, he will not 
be apt to get them so nearly correct when drawing them approximately. 

Problems of this kind are about the best exercises for beginners in mechanical 
drawing. 

The desired results will not be obtained unless extremely fine, hair- like 
lines are drawn, and all centres and other points taken exactly on these lines. 
Therefore it is important to use very hard pencils sharpened to a needle or chisel 
point, and to avoid making holes in the paper with the compasses. 

Given and required lines should be drawn fine in ink, and construction lines 
drawn in pencil only ; but, to distinguish them in these problems, given lines are 
heavy; required lines fine; and construction lines broken. 

An ARC of a circle is drawn by stickmg one point of the compasses, Fig. 15» in 
the paper and moving the other point around on the paper a short distance. The 
distance between the points of the compasses is the radius of the arc. An arc 
can be drawn when its centre and radius are given. 

PROB. 1 . Draw a line which will divide a given line into two equal parte 
and be perpendicular to it. (Take any line.) 

With the ends of the given line as centres, and with equal radii, draw arcs which 
will intersect on both sides of the line. The line joining the points of intersection 
will be the one required. 

PROB. 2. Draw a line perpendicular to a given line through a given point 
in the line. (Take any point in a line.) 

With the given point es a centre, and the same radius, draw arcs intersecting 
the line. With the two jDoints of intersection as centres and equal radii, draw- 
arcs intersecting on both sides of the line. A line joining these two last points 
of intersection will be the perpendicular required. 

PROB. 3. Draw a line perpendicular to a given line from any given point. 
(Take point 2 ins. from line.) 

With the given point as a centre, draw an arc which intersects the 
line at two points. With these two points as centres, and equal radii, 
draw intersecting arcs. The line joining the given point and the last point of 
mtersection will be the perpendicular required. 

An ANGLE is formed by two straight lines meeting in a point. Shcrttningor 
lengthening the lines does Eot change the angle. Fcr the purycse of mrasuriug 
angles and arcs, the whole circumference of anv circle is arbilr; rily divided into 



6 MECHANICAI. DRAWING. 

360 equal arcs, called degrees. If the vertex of an angle (point where the sides 
meet) is placed at the centre of the circle, the arc between the sides will contain 
the same number of degrees as the angle. A right angle contains 90 degrees. 
The radius of a circle will step around on the circumference exactly six times, 
dividing it into arcs of 60 degrees. 

PROB. 4, Draw a line which will divide a given angle, or arc, into two 
■equal parts. (Take any angle.) 

With the vertex of the angle as a centre, draw an arc intersecting 
the sides of the angle. With the points of intersection as centres, and 
equal radii, draw intersecting arcs. The line joining the last point of inter- 
section with the vertex, will bisect the angle and the arc contained between its sides. 

Any point in this line will be equally distant from the sides of the angle. 

PROB. 5. Divide a right angle into three equal parts, thereby constructing 
angles of BO degrees. 

With the vertex of the right angle as a centre, draw an arc intersecting the 
sides. With the points of intersection as centres, and the same radius used in 
drawing the arc, draw arcs intersecting the first arc. Lines drawn from the last 
points of intersection to the vertex will trisect the right angle, forming angles of 
■30 degrees each. 

PROB. 6. Construct an angle at a given point, equal to a given angle. (Take 
any convenient angle and point.) 

With the vertex of the given angle as a centre, draw an arc intersecting 
the sides; and, with the vertex of the required angle as a centre, draw 
an arc with the same radius. Make the two arcs between the sides equal 
and the angles will be equal. 

Lines are PARALLEL when they lie in the same plane and never meet if 
produced indefinitely in both directions. The surface of the paper is the plane 
in which lines are drawn. 

PROB. 7. Draw a line parallel to a given line, at a given distance from 
it. (Take 2 ins.) 

With a radius equal to the given distance, and two points in the line — 
one near each end — as centres, draw arcs. A line just touching these arcs will 
be the parallel line required. 

PROB. 8. Draw a line parallel to a given line, through a given point. (Take 
point about dh ins. from line.) 

Draw a line through the point and crossing the given line. Construct angles 
around the point equal to the corresponding angles formed by the intersection of 
the two lines. A side of these angles will be the parallel line required 

PROB. 9. Divide a given line into any number of equal parts. (Take 7 parts.) 

Draw a line through one end of the given line, and from that end measure 
off the required number of equal divisions on this second line. Draw a line 
through the last point of division and the other end of the given line, and draw 



MECHANICAL DRAWING. 7 

liaes through all the other points of division parallel with this last line. These 
parallel liaes will divide the given line into the required number of equal parts. 

A TRIANGLE is a plain figure bounded by three straight sides. When one 
angle is a right angle the figure is a right triangle; the longest side of which is 
the hypotenuse, and the hypotenuse squared is equal to the sum of the squares of 
the other two sides. When two of the sides are equal, the two angles opposite 
them are equal and the triangle is isosceles. The angles of any triangle added 
make two right angles The area of a triangle is equal to half its base multiplied 
by its height. 

PROB. 10. Form a right angle at the end of a given line by constructing a 
right triangle on the line. 

Measure off four equal divisions from the end of the line. With a radiu s 
equal to 3 of the divisions, and the end of the line as a centre, draw an 
arc. With a radius equal to 5 of the divisions, and the fourth point on the line 
as a centre, draw an arc intersecting the first arc. Draw a line from the point of 
intersection to the end of the given line and it will form a right angle with the line. 

Peoof. — The hypotenuse squared is 25; 3 squared is 9; 4 squared is 10; 9 and 16 
make 25. 

PROB. 11. Construct a right triangle containing two angles of 45 degrees each. 

Construct a right angle, (Prob. 1, 2 or 10.) With the vertex as a 
centre, draw an arc intersecting the sides. Draw a line joining the points of inter- 
section and it will form angles of 45 degrees with the sides. 

PROB. 12. Construct a right triangle containing one angle of 30 degrees 
and another of 60. 

Construct a right angle. With the vertex as a centre and any radius, 
draw an arc intersecting one side. With the point of intersection as a 
centre, and twice the first radius as a radius, draw an arc intersecting the other 
side. Draw a line through the two points of intersection, and it will form the re- 
quired angles with the sides. 

A. POLYGON is a figure bounded by three or more straight lines in the 
same plane. When all the sides are equal, and all the angles are equal, it is a 
regular polygon; and a circle can be drawn around it, touching all the angles, and 
another can be drawn within it, touching all the sides at their middle points. 

PROB. 13. Draw a regular polygon of 5 sides in a circle of a given diameter, 
(Take diameter of circle 2 ins.) 

Draw a diameter of the circle, and a radius perpendicular to it. Take 
a point on the diameter, half way between the centre and the circum- 
ference, as a centre; and, with a radius equal to the distance 
from this point to the extremity of the radius, draw an arc intersecting the diam- 
eter. The distance between this point of intersection and the extremity of the 
radius will be a side of the required pentagon, and it can be completed by step- 
ping the side around on the circumference and drawing lines joining these points. 



'S MECHANICAL DRAWIN©. 

PROB. 14. Draw a regular polygon of 6 sides in a circle. (Take diameter of 
circle 2 ins.) 

The radius of the circle will be the sides of the inscribed hexagon, and 
it can be completed by stepping the radius around oa the circumference and 
drawing lines joining these points. 

PROB. 15. Divide the circumference of a circle into 24 equal parts; thereby 
making arcs and angles of 15 degrees each. (Take diameter of circle 2 ins.) 

Draw two diameters perpendicular to each other. (Prob. 1.) With the radius of 
the circle as a radius and the extremities of the diameters as centres, draw arcs 
intersecting the circumference at 8 points. Bisect the arcs between all these 
points (Prob. 4) and the required number of 'divisions will be made. 

PROB. 16. Draw a regular pentagon with sides of a given length. (Take 
sides 14 ins.) 

Draw a side and erect a perpendicular to this side at one end (Prob. 2 
or 10) equal in length to half the side. (Prob. 1.) From the other end of 
the side draw a line passing through the extremity of the perpendicular, and ex- 
tend this line a distance beyond the perpendicular equal to half the side. The 
distance between the end of this line and the nearer end of the side will be the 
radius of the circumscribed circle. It may be drawn and the polygon completed 
by stepping the given side around on it 5 times. 

PROB. 17. Draw a regular G sided polygon, the distance between the oppo- 
site sides being given. (Take distance 1-| ins.) 

Draw parallel lines the given distance apart (Prob. 7), and draw a 
line making an angle of GO degrees with them. (Prob. 5.) This line between 
the parallel lines will be the diameter of the circumscribed circle, and half of 
it will be equal to a side of the required hexagon. It may be quickly completed 
with a G0° triangle. Fig. 5. 

PROB. 18. Draw a regular 8 sided polygon in a given square. (Take a 
square 2x2 ins.) 

Find the centre of the square by drawing diagonal lines through 
the vertices (corners). With the vertices as centres, draw arcs passing 
through the centre of the square and intersecting the sides. These points of 
intersection will be the corners of the required octagon. 

The circumference of a CIRCLE is drawn by a point moving around another 
point (the centre) and remaining the same distance from it in the same plane. 
A line drawn from the centre to the curve is the radius, and a line drawn across 
the circle through the centre is the diameter. A line drawn across elsewhere is a 
chord, and any part of the curve is an arc. The whole surface within the circum- 
ference is the area, one-half is a semicircle, and one-fourth is a quadrant. The 
surface enclosed by two radii and an arc is a sector, and that enclosed by a chord 
and an arc is a segment. The circumference is equal to about 3 1-7 times the 
diameter, or more exactly, 3.14I592G53590 times. The area of a sector is equal 






Prob.7. 


-■ 


Prcb.8. 




Frob.^. 


__- — - 


/ 


^\ ' i ' 1 1 1 

% / / / 

^ 1 1 1 

\ 1 / 

\' / 

\ 

\ 


-/ 


VrobJO. 

N 

N 

\ 




Prob.//, 


- 


ProbJZ. 















MECHANICAL DRAWING. 9 

to half its arc multiplied by the raiias, and the area of the whole circle is equal 
to half the circumference multiplied by the radius. 

PROB. 19. Draw a straight line, equal in length to the circumference of a. 
given circle. (Take a circle with diam. 2 ins.) 

Draw two diameters perpendicular to each other, and draw a chord from the 
extremity of one to the extremity of the other. Draw a perpendicular to this 
chord through its centre. (Prob. 4.) The part of this perpendicular between 
the chord and the arc, added to three times the diameter, will be very nearly 
equal to the circumference. It will be about 1-208 of the diameter too great. 

Proof. — Take thechord=2; the radius will be\/2 = 1.414, and the part between 
the chord and arc will be .414-=- 2. 828 = .1464 which is slightly greater than .1416. 

The chord is equal to the radius of a circle which has twice the area of the- 
given circle. 

PROB. 20. Draw a square with an area equal to the area of a given circle. 
(Take circle with diam. 2 ins.) 

Draw a line equal in length to half the circumference Ol the given 
circles (Prob. 19) and extend the line a distance equal to the radius. With 
the middle point of the whole line as a centre, draw a semicircle passing through 
its ends. At the point where the two lines meet, erect a perpendicular. (Prob. 2.) 
The part of it between the line and semicircle will be a side of the required square. 

Proof. — The two parts of the diameter multiplied together equal the perpen- 
dicular squared; as it is a mean proportional between them. (See Wentworth's. 
Geometry, page 157.) 

A circle with an area two, three, four, five or more times as great as the area of 
a given circle may be drawn by making the shorter part of a diameter equal to 
the radius of the given circle and the longer part equal respectively to two, three, 
four, five or more times that radius. The perpendicular to the diameter, from the 
point where the two lines meet to the circumference, will be equal to the radius of 
the required circle. 

Proof. — Let r=: given radius; p= perpendicular, and ar the longer part of the 
diameter. Then ar^ = p^-, r23.1416 = given area; p23.1416 = required area= 
ar^d.14:lQ^^a times the given area. 

PROB. 21. Draw a circle equal in area to a given equare. (Take side of 
square ]f inches.) 

Draw a line from a vertex of the square to the centre of oae 
side. This line will be very nearly equal to the diameter of the required circle. 
It will be about 1-100 of the side too small. 

Proof. — Take side of square=l ; the diameter of circle will be 1.118, but it should 
be 1.128 to make the areas exactly the same. 

PROB. 22. Draw a circle passing through 3 given points. (Take points 1, 1|, 
and 2 ins. apart.) 

With two of the given points as centres, and with equal radii, 
draw arcs which will intersect at two points. Draw a line through these two points 
of intersection. With the third given point and either one of the others as centres. 



10 MECHANICAI, DRAWING. 

draw arcs and another line as before. The point where these two lines 
cross will be the centre of the required circle. 

By taking any three points in the circumference of a given circle or arc, its 
centre may be found in the same way. 

PROS. 23. Draw, mechanically, an arc passing through 3 given points, with- 
out using a centre. (Take points 1^, Ig and 2 |- ins. apart.) 

Locate the three points on a piece of firm card-board, and cut straight lines passing 
from the intermediate point through the two extreme points. Stirkfine pins in the two 
extreme points on the drawing paper, aod hold a pencil in the vertex of the angle formed 
in the card-board while it is moved back and forth against the pins. The pencil 
will draw the required arc, and it can be inked with a curved ruler, Fig. 16. 

PROB. 24-. Draw an arc with a given radius without using a centre. (Take 
radius G inches.) 

Part of a regular polygon of twelve sides can first be drawn, and the 
arc drawn through three or more of the corners. (Prob. 23.) The angles at the 
corners will be 150 degrees (Prob. 5); and the sides can be found by constructing 
a triangle with the base equal to half the given radius, and the angles at the base 
90 and 15 degrees. (Prob. 12 and 4.) The hypotenuse will be the required side. 
If the radius is quite large, any regular polygon may be taken. The angles can 
bo easily computed and the sides found by a table of chords. 

A line is TANGENT io a circle when, however far produced, it passes through 
hut one jyoint in the circumference of the circle. It will be perpendicular to a 
radius of the circle drawn to the point of tangency. 

PROB. 25. Draw a line tangent to an arc at a given point in the arc. (Take 
radius of arc 5 ins.) 

With the given point as a centre, draw arcs intersecting the 
given arc at two points equally distant from the given point. "With these two 
points as centres, draw intersecting arcs, and draw lines through the points of 
intersection. (Prob. 22.) A line drawn perpendicular to this line through the given 
point (Prob. 2) will be the tangent required. 

PROB. 26. Draw a line tangent to a circle from a given point outside of 
the citcle. (Take circle 2 ins. in diam.. point 3 ins. from centre.) 
With the given point as a centre, draw an arc which will pass through the centre of the 
circle. With the centre of the circle as a centre, and with a radius equal to the diameter 
of the circle, draw an arc intersecting the first arc. Draw a line from the last point 
of intersection to the centre of the circle. This line will cross the circumference 
at the point of tangency; and a line drawn through it from the given point will be 
the tangent required. Two tangents can be drawn. 

PROB. 2T. Draw a line tangent to two circles of different diameters. (Take 
circles 1 and 2 ins. in diam., centres 2 ins. apart.) 

Draw a line through the centres, and extend it beyond the 
smaller circle. Draw parallel lines through the centres of the circles. 



ProbJ3 




ProbJ4 




Prob./e 




Proh.l? 






J>TobJ8 



A 




•:>-] 




.X 








\/ 



FTob. /£) 




JProh^O 



JProb.2/ 




Prob.£^ 




Prob.£J 




Prob.£' 




MKCHANICAL DRAWING. 11 

Through the points where these parallel lines intersect the circumferences 
draw a line, and extend it until it crosses the line passing through 
the centres. From this last point of intersection draw a tangent to either circle. 
(Prob. 26.) It will be the tangent required. Four such tangents can be drawn; 
two will be interior, and two exterior tangents. 

PROB. 28. Draw a circle tangent to a line at a given point, and passing 
through another given point. (Take one point 1 in. from the line and 1| ins. 
from the point of tangency.) 

Draw a perpendicular to the line at the given point in it. 
(Prob. 2.) With the points as centres and with equal radius, draw arcs inter- 
secting at two places, and draw a line through the points of intersection. 
(Prob. 22.) It will cross the perpendicular at the centre of the required circle. 

PROB. 29. Draw a circle tangent to two lines and passing through a given 
point equally distant from them. (Take lines making an angle of 60 degrees, and 
a point 1 in. from vertex.) 

Extend the lines until they meet, and draw a line 
bisecting the angle between them (Prob. 4) Draw a perpendicular to this line 
through the given point (Prob. 2), and bisect the angle which the perpendicular 
makes with one of the sides. (Prob. 4,) The two bisecting lines will cross at the 
centre of the required circle. 

Note. — If the lines cannot be made to meet on the paper, draw lines parallel to and 
equally distant from them (Prob. 7) and bisect the angle formed by these lines. 

PROB. 30. Draw a circle tangent to three given lines. (Take one line 3 ins. 
long, and the others making angles of 45 and 60 degrees at its ends.) 

Extend the lines until they meet, and draw lines bisecting any two of the angles 
formed. (Prob. 4.) These bisecting lines will cross at the centre of the required circle. 
Its radius can be found by drawing a perpendicular to one of the given lines. 
(Prob. 3.) 

Two CIRCLES ARE TANGENT when the circumference of one passes 
through but one point in the circumference of the other. A line drawn through 
the centre of two tangent circles will pass through the point of tangency. 

PROB. 31. Draw a circle with a given radius, tangent to a line and a given 
circle. (Take radius of given circle 1 in., with centre 1^ ins. from line; radius 
of required circle, | ins.) 

Draw a line parallel with the given line at a distance from it 
equal to the given radius (Prob. 7); and, with the centre of the given circle 
as a centre, and a radius equal to the radius of the given cii'cle added to the 
radius of the required circle, strike an arc. It will cross the parallel line at the 
centre of the required circle. 

PROB. 32. Draw a circle tangent to a circle and a line at a given point in 
the line. (Take line and circle, as in Prob 31, and point in line 2 ins. from centre 
of given circle.) 

Draw a p3rp3niicular to the line through the given point 



12 MECHANICAI, DRAWING. 

(Prob. 2) and extend it on either side of the line a distance equal to the radius of 
the given circle. Draw a line from the extremity of the perpendicular to the 
centre of the circle, and draw a perpendicular to this line through its middle 
point. (Prob. 1.) It will cross the first perpendicular at the centre of the required 
circle. Two such circles can be drawn; the given circle will be tangent to the ex- 
terior of one and to the interior of the other. 

PROB. 33. Draw a circle tangent to a line and to a circle at a given point 
in the circle. (Take line and circle, as in Prob. 31, and point in circle 1 in. from 
the line.) 

Draw a line from the centre of the circle through the given point, and 
draw another through the point tangent to the circle. (Prob. 25.) Extend the 
tangent till it meets the given line, and draw a line bisecting the angle between 
them. (Prob. 4.) The bisecting line will cross the first line at the centre of the 
required circle. Two such circles can be drawn; the given circle will be tangent 
to the exterior of one and to the interior of the other. 

PROB. 34. Draw a circle with a given radius, tangent to two given circles. 
(Take given circles 1 and 2 ins. in diameter, and centres 2 ins. apart ; radius of 
required circle | ins.) 

With the centre of one circle as a centre, and with a radius 
equal to its radius added to the radius of the required circle, draw an arc. With 
the centre of the other circle as a centre, and with a radius equal to its radius 
added to the radius of the i-equired circle, draw another ai-c. The two arcs will 
cross at the centre of the required circle. 

PROB. 35. Draw a circle tangent to two given circles at a given point in one 
circle. (Take given circles as in Prob. 34, and point in larger circle 30 degrees from 
aline between their centres.) 

Draw a line through the given point and the centre of that cii'cle. 
With the centre of that circle as a centre, draw three or four arcs 
crossing this line near the centre of the required circle. With the centre of the 
other circle as a centre, and a radius equal to its radius added to the distance 
from the given point to the first arc, draw an arc intersecting the first arc; then 
with the same centre and the same radius increased by the distance from the first 
arc to the second, draw an arc intersecting the second arc; then in the same way 
draw one intersecting the third arc and so on. With a curved ruler, Fig IG, 
draw a curved line through these points of intersection. It will cross the first line at 
the coutre of the required circle. Tae curve is part of a hyperbola (Prob. 44) and 
curves towards the smaller circle. 

PROB. 3G. Draw a circle tangent to three given circles of different diam-ters. 
(Take centres of circles all 2 ins. apart; with radii of f, | and 1 inch.) 

Draw two curves as in Prob. 35, and they will cross each other at the centre of the 
required circle. 

An ELLIPSE is drawn by a point moving around iwo other points in the same 



Prck25. 




Proh26. 





ProbZa, 




Pmh.29. ^ 




ProhSO. 




ProhJl 




ProKJ2. 




Proh.33, 




PwhJ4. 




Prch3§. 




Proh36. 




MECHANICAL DRAWING. IS 

plane, so that the distance between it and one point, added to the distance between 
it and the other point, remains the same. The two fixed points are the focii. A 
line drawn through the focii is the longer axis of the ellipse, and a perpendicular 
to it through its middle point is the shorter axif^. 

PROB. 37. Draw an ellipse with pins and a string; the axes being given. 
(Take axes 1^ and 2| ins.) 

Draw the shorter axis. With its ends as a centres, and a radius 
equal to half the longer axis, draw arcs intersectiDg at two points. These 
points will be the focii. Stick pins in the focii and in one end of the shorter axis, 
and tie a linen thread around the three pins. Remove the pin from the shorter 
axis. The point of a pencil held tight against the string will draw the required 
ellipse. 

The position of the focii and the length of the string can be computed: 
The distance from the point where the axes cross to the focii will be equal to the 
base of a right triangle; the altitude of which will be half the shorter axis, and 
the hypotenuse half the longer axis. The string will be equal in length to the 
longer axis added to the distance between the focii- 

PROB. 38. Draw an ellipse by taking 3 points on a straight edge; the axes 
being given. (Take axis 1^ and 2|^ ins.) 

Draw the axes perpendicular to each other through their middle points. Take 
3 points on the straight edge of a piece of paper or a scale, Fig. 17, and make the 
distance between the first and second points equal to half the shorter axis; and, 
between the first and third, equal to half the longer axis. Keep the second point 
on the longer axis, and the third point on the shorter axis. The first point will 
draw the required ellipse. 

The straight edges of a square may be held on part of the axes, while a thin 
piece of wood, with pins in it for guides, is used for drawing the curve; or an 
instrument called a trammel may be used for drawing large ellipses, and an 
ellipsograph used for drawing small ones. 

PROB. 39. Draw an ellipse approximately with 4 arcs; the axes being given, 
(li and 2| ins.) 

Draw the axes as in Prob. 38 and from one end of the longer axis measure a 
distance equal to halt the shorter axis. Construct an angle of 45 degrees at this 
point meeting the shorter axes (Prob. 11). With this point as a centre, and half 
the hypotenuse of the triangle formed, as a radius, strike an arc intersecting the 
longer axis. With the point where the axes cross as a centre, and a radius equal 
to the distance from it to the further point of incersection, draw arcs crossing the 
axes at 4 points. Draw lines from the two points in the shorter axis through 
the two in the longer axis and extend them. These four points will be the centres, 
and the arcs will be drawn through the extremities of the axis, between the ex- 
tended lines. 

PROB. 40. Draw an ellipse approximately with 8 arcs; the axes being 
given. (1^ and 2 J- ins.) 



14 MECHANICAL DRAWING. 

Eight centres are used; two on each axis, and one in each angle between 
them. Draw the two axes as in Prob. 38. From an extremity of each axis 
draw a line, 1 and 2, parallel with the other axis and forming a rectangle 
on half the axes. Draw line 3 from the extremity of one axis to that of the 
other, diagonally across the rectangle. From the outside corner of the rectangle 
draw line 4 perpendicular to the diagonal and it will cross the axes at two of the 
required centers, a and b. With the intersection of the axes as a centre, and half 
the shorter axis as a radius, strike an arc intersecting the longer axis at 5. With 
a point half way between this point and the farther extremity of the longer axis, 
as a centre, draw a semicircle 6 passing through these two points. With the radius 
of this semicircle as a radius and the extremity of the shorter axis as a centre, 
strike an arc intersecting the shorter axis. With the centre b on the shorter axis 
extended, as a centre, draw arc 7 passing through the point of intersection oq the 
shorter axis. With the extremity of the longer axis as a centre, and a radius equal 
to the length of the shorter axis between the centre and semicircle, as a radius, 
draw arc 8. It will intersect the last arc drawn at another one of the required 
centres c. Draw line 9 through the centres b and c, and line 10 through c and a. 
Three of the arcs will be drawn between these lines extended for one-fourth of the 
ellipse. The other centres and lines can be easily found from these. 

PROB. 41. Draw a perpendicular and tangent to an ellipse at a given point 
on the curve. (Take axes 1^ and 2 J ins. and a point about equally distant from the 
ends of the axes.) 

Draw a line from both focii through the given point, and bisect the angles they 
form (Prob. 4). One of the bisect'ng lines will be the required perpendicular, 
and the other the required tangent. 

PROB. 42. Find the two axes of an ellipse; the curve only being given. 
(Take same sized ellipse.) 

Draw any two parallel lines across the ellipse, and draw a line through the 
middle points of these lines. With the middle point of this line as a centre, draw 
a circle intersecting the curve at four points. With these four points as centres 
and equal radii, draw intersecting arcs and draw lines through the points of inter- 
section. These lines will be the required axes. 

A PARABOLA is drawn by a point moving in a plane, so that its distance from a 
given point remains equal to its distance from a given line. The fixed line is the 
focus and the line is the directrix. 

PROB. 43. Draw a parabola by finding points in the curve. (Take focus 
^ in. from directrix.) 

Draw lines parallel with the directrix (use T-square, Fig. 4.) With the 
distance from the directrix to the first parallel line as a radius, and the 
focus as a centre, draw an arc intersecting the first parallel line at two points. 
With the distance from the directrix to the second parallel line as a radius, and 
the focus as a centre, draw an arc intersecting the second parallel line. In the 



MECHANICAI, DRAWING. 15 

same way draw arcs intersecting the other parallel lines. These points of inter- 
section will be points in the required curve. It may be drawn in iak, free hand or 
with a curved ruler, Fig. 10, or with the compasses by finding centres and 
radii by trial, which will draw arcs through three of the points at a time. 

A HYPERBOLA is drawn by a point moving in a plane, so that its distance 
from a given point remains equal to its distance from a given circle. The fixed 
point and the centre of the circle are the focii. 

PROB. 44, Draw a hyperbola by finding points in the curve. (Take radius 
of circle 1 in. ; focii 1^ ins. apart.) 

With the centre of the circle as a centre, draw a number of arcs 
where the required curve is to be drawn. With the distance from the 
circle to the first arc as a radius, and the other focus as a centre, 
draw an arc intersecting the first arc at two points. With the distance from the 
circle to the second arc as a radius, and the same focus as a centre, draw an arc 
intersecting the second arc. In the same way draw arcs intersecting the other 
arcs. These points of intersection will be points in the required curve, and it 
may be drawn in ink in the same way as the parabola. 

A HELIX is generated by a point moving uniformly around a given line, and 
also moving in the direction of the line at a fixed distance from it. The given 
line is the axis. A corkscrew, a wire spring, and screw-threads are illustrations 
of a helix. Two views are necessary in a drawing to show a helix. The bottom 
view will be a circle and the side view will be reversed curves. 

PROB. 45. Draw a tielix with a given diameter and a given rise per revo- 
lution. (Take diameter 2 ins. and rise 1 in. per revolution.) 

Draw a circle for the bottom view. From the centre of this circle draw the 
axis for the side view. Measure the rise per revolution on this axis. Divide the 
rise into 24 equal parts (Prob. 9 or with Scale, Fig. 17), and draw lines through 
these points of division perpendicular to the axis (use T-square). Divide half the 
circle unto 12 equal parts (Prob. 15). From the first point of division on the 
circle, draw a line parallel with the axis (use a triangle. Fig. 5) and intersecting 
the first line which is perpendicular to the axis; from the second point of division 
on the circle draw another parallel line intersecting the second perpendicular line; 
from the third draw one intersecting the third, and so on. These points of inter- 
section will be points in the required curve. The curve may be traced on a piece 
of firm card-board or thin wood, and trimmed out smoothly with a keen pen-knife. 
With this curve a great many revolutions of the helix may be made neatly with 
ink in the drawing. 

A SPIRAL is drawn by a point moving uniformly around a given point in the 
same plane, and moving away from it at the same time. A watch spring is a^ 
illustration of a spiral. This curve is used for drawing cams. 

PROB. 46. Draw a spiral moving uniformly from the centre at given rate. 
(Take 1 in. per revolution.) 



16 MECHANICAL DRAWING. 

Draw lines through the centre, making angles of 15 degrees (Prob. 15). Meas- 
ure the distance per revolution on one of these lines, and divide the distance into 
24 equal parts. With the point as a centre and a radius equal to the distance to 
the ^rs^ point of division, draw an arc intersecting one of the lines; with a radius 
equal to the distance to the second point of division, draw an arc intersecting the 
n<ext line; from the third point of division draw an arc intersecting the third line, 
and so on. These points of division will be points in the required spiral. It may- 
be drawn in ink with the compasses by finding centres and radii by trial, which will 
draw arcs through three of the points at a time. 

The INVOLUTE of a circle is drawn by a point in a straight line which rolls 
on the circle. A pencil fastened to a string, which is kept stretched as it is un- 
wound from a spool, -vrill draw an involute of a circle. This curve is used for 
drawing the teeth of gear-wheels. 

PROB. 47. Draw the involute of a given circle. (Take circle 1 in. in 
diameter.) 

Divide the circle into 24 equal parts. (Prob. 15), and draw lines tangent to the 
circle at these points of division (with triangles. Fig. 5). With one point of divis- 
ion as a centre, and with a radius equal to the length of the arc between the points 
of division, draw an arc from the circle to the first tangent line; with the next point 
of division as a centre, draw an arc from the end of this arc to the next tangent 
line; with the third point as a centre, continue the curve to the third tangent line, 
and so on. When 24 divisions are taken the curve will be more accurate, if these 
■centres are taken on a circle whose diameter is about 1-94 greater than the diam- 
eter of the given circle. 

A CYCLOID is drawn by a point in the circumference of a circle which rolls 
on a line. If the generating circle rolls on the outside of a circle, the curve is an 
epicycloid ; and, if it rolls on the inside, it is a hypocycloid. The circle on which 
it rolls is the pitch circle. These curves are used for drawing the teeth of gear- 
wheels. 

PROB. 43. Draw epi- and hypocj^cloids; the diameters of the circles being 
given. (Take diameter of pitch circle 4 ins., and diameter of generating circles 1 
in., and let both curves start from the same point.) 

Draw part of the pitch circle, and draw a generating circle tangent to it on the 
■outside, and draw another tangent on the inside. With the centre of the pitch 
circle as a centre, draw arcs passing through the centres of the generating circles. 
Make a number of equal divisions on the pitch circle, and draw lines through 
them from the centre and intersecting the two arcs. With these points of inter- 
section on the arcs as centres, draw parts of the generating circle where the curve 
is to be drawn. With a point of division on the pitch circle as a centre, and the 
distance between the points of division as a radius, draw an arc from the pitch 
circle to the nearest pari of the generating circle; with the next point of division 
as a centre, draw an arc which will continue the curve to the next part of the gen- 



ProhJZ 




£Ciip6 



ProhSS. 




PrchM 




Proh40. 




ProA41. 




Proh4Z. 





MKCHANICAI. DRAWING. 17 

erating circle; with the third point as a centre, continue the curve to the third part 
of the generating circle, and so on. Unless the points of division are quite small, 
the radii of the arcs will be slightly too great. 



I 



CHAPTER III. 

PROJECTIOIS^ AND DEVELOPMENT OF SOLIDS. 

The problems in this chapter are given to show how the different 
views of an object are arranged and how patterns for curved surfaces are 
cut. Two views are usually sufficient to represent the dimensions of a 

solid. The top views, when drawn, are placed exactly above the prin- 

cipal views; the bottom views, exactly below them; and the side and end 
views, near the sides and ends which they represent. 

In the figures, the outlines are drawn heavy on the bottom and right 
to shade them; and surfaces, which are cut by a plane, are distinguished 
by having oblique parallel lines drawn across them, called section-lines or 
hatching. 

A good drawing board, T-square, triangle, and dividers or measuring 
scale, are essential for making these and similar drawings. The student 

should copy the developments on heavy paper and cut them out and bend 
them into shape to test the accuracy of his drawings. 

A point is PROJEGTED on a line or a plane when a straight 
line is drawn from it to the line or plane. The line drawn is the pro- 

jecting line, and the point where the projecting line intersects the other 
line or pierces the plane is the projection of the given point on that line 
or plane. 

A surface is DEVELOPED when it is removed from a solid 
and spead out on a plane. 

A PRISM is a solid with two equal parallel bases which are poly- 
gons, and with faces (sides) which are all parallel to the same line. 
When the bases are regular polygons, the prism is regular; and when the 
faces are perpendicular to the bases, it is a right prism. 

PRO-B. A 9. Draw three views and the development of the 

faces of a regular 6-sided prism with its top cut away by a plane which 
makes an angle of 45 degrees with its base. (Take prism i inch across 
corners and 2 ins. high.) 

Draw a regular hexagon for the bottom view. Project vertical lines 

upwards from the corners of the bottom view, as indicated in the figure 
by dotted lines with arrow points. These projecting lines determine the 

edges in the view which is above. Horizontal lines projected to the 

right from the corners determine the edges in the view which is at the 
right of the bottom view. At any convenient distance from the bottom 

view, draw straight lines across the projecting lines perpendicular to them, 
for the base in the other two views. Draw a line across the upper view 



6 MECHANICAL DRAWING. 

at the required distance from the base and making the required angle with 
it. This' last line represents the cutting plane. It crosses all the 

edges of the prism, and the length of each edge may be measured, and 
the corresponding edges in the view at the right made the same length. 

The development of the faces may be drawn by drawing parallel 
lines, as far apart as the edges really are on the prism itself, and making 
each line equal in length to the corresponding edge of the prism. The 

distance between the parallel lines is equal to the distance between the 
corners in the bottom view. The length of each one may be projected 

from the upper view, as indicated in the figure by dotted lines. 

PROB. SO. Draw three views and the development of the 

faces of a regular 8-sided prism with its bottom cut away by a plane 
which makes an angle of 30 degrees with its base. (Take prism i inch 
across corners and 2 ins. high.) 

Draw a regular octagon for the top view. Project lines downwards 

from the corners and they will determine the edges in the view which is 
below it. Draw a line across this view making the required angle with 

the base. It will cut all the edges, and the length of each edge may 

be measured and the view at the right and the development of the faces 
drawn as in Prob. 49. 

PROB. 51. Draw three views and the development of the 

faces of a regular 24-sided prism with its top cut away by a plane which 
makes an angle of 60 degrees with its base. (Take prism i inch across 
corners and 2 ins. high.) 

Draw a regular polygon of 24 sides for the bottom view. Project 

lines from its corners for the edges in the other two views. Draw a 

line across the upper view, making the required angle with its base, and 
complete the other view and the development as in Prob. 49. 

A CYLINDER is a solid with two equal, parallel bases which 
are circles or other plane curves, and with a curved lateral surface which 
is generated by a straight line moving parallel with itself and constantly 
touching the curves. Any position of the generating line is an element 

of the cylinder. The line joining the centres of the bases of a regular 

prism or a cylinder is its axis. 

The volume of a prism or cylinder is equal to its base multiplied by 
its altitude (height). 

PROB. 52. Draw three views and the development of the 

whole surface of a circular cylinder with its bottom cut away by a plane 
which makes an angle of 45 degrees with its axis. (Take cylinder i 

inch in diam. and 2 ins. high.) 

Draw a circle for the top view and divide it into 24, 48, or any num- 
ber of equal parts. Project lines drawn from these points of division 



MECHANICAL, DRAWING. 7 

and they will be elements of the cylinder in the lower view, which are 
equally distant from each other. Draw the base and draw a line across 

the lower view making the required angle with its axis. It will cut all 

the elements, and the length of each one may be measured, and the oth- 
er view and the development of the surface drawn as in Prob. 51, with 
the exception that the width across the development is equal to the cir- 
cumference of the base, and the outline is a curve traced through the ex- 
tremities of the elements. The outline of the surface which is cut by 
the plane is known to be an ellipse. Its shorter axis is equal to the 
diameter of the cylinder and its longer axis is equal to the line drawn 
across the lower view. 

A PYRAMID is a solid with one base, and triangular /«,f(?j- which 
meet at a point called the apex or vertex. When the base is a regular 

polygon and the faces are equal, the pyramid is 7-egular. 

PROB. 53. Draw three views and the development of the 

faces of a regular 6-sided pyramid with its top cut away by a plane 
which makes an angle of 45 degrees with its base. (Take pyramid i^ 

ins. across corners of base and 2 ins. high with y^ inch of top cut off.) 

Draw a regular hexagon for the outline of the base in the top view, 
and draw a horizontal line at a convenient distance below it for the base 
in the view below it. Project the corners of the base from the top 

view down to the base in the lower view. Locate the apex in the low- 

er view at the required distance above the base and exactly below the 
centre of the top view. Draw lines from the corners of the base to the 

apex in both views. These lines are the edges of the pyramid. Draw 

a line across the lower view making the required angle with the base. 
It will cut all the edges. Project the points where they are cut up to 

the corresponding edges in the top view, as indicated in the figure by dot- 
ted lines. Draw lines joining these points on the edges in the top view, 
and its outline will be completed. The other view may be projected to 

the right of the lower view. The distance across its base is equal to 

the vertical distance across the top view; and the base, apex, and points 
where the edges are cut, can be projected from the lower view, as indi- 
cated in the figure by dotted lines. 

The development of the faces of the complete pyramid is 6 equal 
isosceles triangles. Their bases are equal to the distance between the 

corners of the base of the pyramid in the top view, and their sides are 
equal to the true length of the edges of the pyramid. The development 
of the faces which are partly cut away may be drawn by finding the 
length of the remaining part of each edge. The edges on the right and 

left, in the lower view to the left in the figure, are parallel with the sur- 
face of the paper and are therefore drawn in their true lengths; but the 



8 MECHANICAL DRAWING. 

other edges come towards the front at one end and are therefore shorter 
on the drawing than they really are on the pyramid. Their true lengths 
may be found by projecting their extremities to the line at the right or 
left, as shown in the figure by the dotted lines with arrow-points. The 

true length of the remaining part of each edge may be measured off on 
the corresponding edge of the development of the complete pyramid, and 
the required development completed as in Prob. 49. 

PROB. 54. Draw three views and the development of the 

faces of a regular 8-sided pyramid with its bottom cut away by a plane 
which makes an angle of 30 degrees with its base. (Take pyramid ij^ 

ins. across corners of base and 2 ins. high. ) 

Draw a regular octagon for the outline of the base in the top view 
of the complete pyramid, and draw a horizontal line below it for the base 
in the side view below it. Draw the corners, as in Prob. 53, and draw 

a line across the side view making the required angle with the base. 
This line will cut all the edges, and the points where they are cut may 
be projected up to the corresponding edges in the top view. The view, 

at the right and the development of the faces may be drawn as in Prob. 

53- 

A CON EI is a solid with one base which is a circle or other plane 
curve, and with a curved lateral surface generated by a straight 
line' passing through a fixed point and constantly touching the curve. 
Any position of the generating line is an element of the cone. The line 

joining the centre of the base with theHapex of a regular prism or a cone 
is its axis. 

The volume of a pyramid or cone is equal to its base multiplied by 
one-third of its altitude. 

PROB. 55. Draw two views and the development of the sur- 

face of a right circular cone, which has its top cut away by a plane par- 
allel with its base. (Take cone 2^ ins. across base and i in. high 
with Yz in. of top cut off. ) 

Draw a circle for the outline of the base in the top view, and draw 
a horizontal line below it equal in length to the diameter of the circle, 
for the base in the lower view. Locate the apex at the required dis- 

tance above the base in the lower view, and draw elements to it from the 
extremities of the base. Draw a line across the lower view parallel 

with the base and cutting away the required amount of the top. The 

outline of the cut surface in the top view is a circle with its diameter 
equal to the line drawn across the lower view. 

The development of the surface of a cone is a sector of a circle Avhose 
radius is equal to an element of the cone, and the arc is equal in length 
to the circumference of the base of the cone. The part cut away from 



Trob.49 



O 



Prism. 




Proh.SO. 




Pr6b.J2. 




cylinder 




MECHANICAL DRAWING. 9 

the development is drawn with the same centre and with a radius equal to 
an element of the part which is cut away from the cone. 

When the elements of a cone form angles of 60 degrees at its base 
and apex, the development is a semicircle. 

Note.— This problem is used in cutting out flanges and such tin utensils as funnels, dippers, 
strainers, pails, coffee-pots, etc. 

PROB. 56. Draw two views and the development of the 

whole surface of a cone which has its top cut away by a plane parallel 
with its base, and its bottom cut away by a plane which makes an angle 
of 30 degrees with its base. (Take cone 2 ins. across base and 2 ins. 

high with ^ in. of top cut off.) 

Draw two views and the development of the whole cone as in Prob. 
55. Divide the circle in the top view into 24, 48, or any number of 

equal parts and project these points of division down to the base in the 
lower view. From these points draw lines to the apex in both views. 

These lines are equally distant elements of the cone. Draw a horizon- 

tal line cutting off the top and an oblique line cutting off the bottom in 
the lower view. The two. lines cut all the elements and the length of 

each one may be found and the development completed, as in Probs. 54 
and ^S, by tracing a curve through the extremities of the elements. 

The surface of the cone which is cut by the oblique plane is known 
to be an ellipse. Its longer axis is equal to the oblique line which is 

drawn across this cone, and its shorter axis is equal to the diameter of 
the cone at the middle of the oblique line, as indicated in the figure by 
the broken horizontal line. 

Solids INXERSEGX when one pierces the other, or when they 
are cut so as to fit each other or become united. If they only touch 

they are tangent but not intersecting. 

PROB. 57. Draw three views and the development of the 

whole surface of a cylinder intersected by a larger cylinder which is per- 
pendicular to it. (Take diam. of one cylinder i in. and the other i^ 
ins.) 

Draw three views of the smaller cylinder as in Prob. 52. In the 

lower view, draw a circle for the end view of the larger cylinder. Diraw 
equally distant elements on the smaller cylinder, as in Prob. 52. These 

elements are all intersected by the larger circle, and the length of each 
one may be measured and the other two views and the development of 
the lateral surface completed as shown in the figure. 

The development of the surface, which fits against the larger cylinder, 
is eliptical, and points in its outline may be found thus: Draw a straight 
line through the centre of the top view of the smaller cylinder and extend 
it. Take any convenient part of this extended line arid make it equal 



10 MECHANICAL DRAWING. 

in length to the part of the larger circle, in the lower view, against which 
the smaller cylinder fits. Make divisions on it equal to the correspond- 

ing divisions which are made on that part of the larger circle by the ele- 
ments of the smaller cylinder. Draw perpendicular lines through these 
points of division. Project the nearest points of division, which are on 
the top view of the smaller cylinder, to the nearest perpendicular line; 
the next-nearest to the next line, and so on. The outline may be traced 
through these points of intersection. 

PROB. 58. Draw three views and the development of the 

lateral surface of a cylinder intersected obliquely by a larger cylinder. 
(Take diam. of one cylinder i in. and the other i}i ins. with their axes 
making an angle of 60 degrees with each other.) 

Draw three views of the smaller cylinder and draw equally distant el- 
ements on it as in Prob. 52. Draw the larger cylinder making the re- 
quired angle with the smaller one in the lower view. The elements 
drawn on the smaller cylinder are intersected by elements of the larger 
cylinder, and these elements and the points of intersection may be found 
thus: Draw arcs, in the upper and lower views, with radii equal to the 
radius of the larger cylinder, and their centres on the axis of the larger 
cylinder. In the top view, draw elements of the larger cylinder through 
the points of division on the top view of the smaller cylinder, and extend 
them to the arc. Measure off these points of division, which are made 
on the arc in the upper view, on the corresponding part of the arc in the 
lower view; and draw elements of the larger cylinder through them. 
These elements will intersect the elements drawn on the smaller cylinder. 
The same elements, which intersect in the top view, intersect in the low- 
er one, and these points of intersection determine the length of each ele- 
ment of the smaller cylinder. The third view and the development may 
be drawn as in Prob. 57. 

PROB. S9. Draw three views and the development of the 

surface of a regular 6-sided prism intersected by a cone whose axis coin- 
cides with the axis of the prism. (Take prism i in. across corners, and 
elements of cone forming an angle of 60 degrees at its apex. ) 

Draw three views of a regular 6-sided prism, as in Prob. 49. Lo- 

cate the apex of the cone on the axis of the prism at the same distance 
from the base in both the side views. From these points draw elements 
of the cone forming equal angles with the axis of the prism. They in- 

tersect the edges of the prism in one view, and the middle of the faces in 
the other view. As all the edges are the same length, and all the faces 

go up the same distance in the middle, their extremities may be located 
and curves drawn through them, as shown in the figure. These curves 



MECHANICAL DRAWING. 11 

are known to be hyperbolas (Prob. 44) but they are usually drawn as 
arcs of circles. 
Note.— This problem is given to show how the chamfer is drawn on nuts and bolt-heads. 

A POLYHEDRON is a solid with four or more faces. When 
the faces are equal, regular, polygons and the vertices are equal, the poly- 
hedron is regular. There are but five regular polyhedrons. Three of 
them have respectively four, eight, and twenty faces, which are triangles; 
one (the cube) has six faces, which are squares; and one has twelve 
faces which are pentagons. 

PROB. 60. Draw three views and the development of the 

faces of a regular polyhedron of 20 sides intersected by a square prism. 
(Take edges of polyhedron y% ins. and prism ^ ins. square. ) 

This problem is intended as an exercise which will give the student 
a clearer understanding of how the different views of an object are pro- 
jected and arranged. He should first make the polyhedron by drawing 
the development, shown in the figure, on card-board, and cutting through 
the full lines and half through the dotted lines. The edges may then 
be brought together and held in place by pasting strips of paper over 
them. He can then study the problem out for himself. 

Any point in the top view must be exactly above the same point in 
the view below it; and any point in one side view must be exactly to the 
right or left of the same point in the other side view. A side view is 

placed opposite and near the side it represents. 

A SPHERE is a solid with a curved surface, every point of 
which is equally distant from the ce?itre. The distance between the sur- 

face and the centre is the radius of the sphere; and the distance through 
the centre is the diameter. 

A zone is part of the surface of a sphere between two circles which 
lie in parallel planes. A lune is part of the surface between two semi- 

circles which meet at the extremities of a diameter. 

The surface of a sphere is equal to its diameter multiplied by its cir- 
cumference; and the volume of a sphere is equal to its surface multiplied 
by one-third its radius. 

PROB. 61. Draw three views and the development of the 

surface of a regular 6-sided prism intersected by a sphere. (Take 

sphere i^ ins. in diam. and prism i inch across corners with its axis 
passing through the centre of the sphere.) 

The outline of the sphere in each view is a circle. At the centre 

B|v~ of the top view draw a regular hexagon for the top view of the prism. 
^1^ Project lines from the corners of the hexagon to the other two views of 
^^Hthe sphere. Draw the top base of the prism on each of the side views 



12 MECHANICAL DRAWING. 

sphere intersects the edges of the prism in one view, and the middle of 
the faces in the other. These points of intersection may be measured 

from the base and curves drawn through them as in Prob. 59. 

PROB. 62. Draw three views and the development of the 

surface of a cylinder intersected by a sphere. (Take sphere i^ ins. in 

diam. and cylinder i in. with its axis passing 3-16 ins. from centre of 
sphere. ) 

Draw three views of the sphere as in Prob. 61. Draw a circle in 

the top view, for the top view of the cylinder. Place the centre of this 

circle at the required distance to the right or left of the centre of the 
sphere. Draw the other two views of the cylinder and divide it into 

equally distant elements, as in the previous problems. Draw horizontal 

lines on the top view of the sphere through the points of division on the 
top view of the cylinder. These lines will be semicircles in the view 

below the top view, and will intersect the same elements which they in- 
tersect in the top view. These last points of intersection determine the 
length of each element, and the other view and the development may be 
completed as in the previous problems. 

PROB. 63. Draw the development of the surface of a 

sphere, approximately, by dividing it into lunes. (Take sphere i^ ins. 

in diam. and divide surface into 12 lunes.) 

The length of a lune is equal to one-half the circumference of the 
sphere, and the width of it at its centre is equal to one of the parts into 
which the circumference is divided. 

Draw a line equal in length to the circumference of the sphere and 
divide it into the required number of equal parts. Draw a perpendicu- 

lar line half way between two of these points of division and extend it 
on both sides a distance equal to one-fourth the circumference of the 
sphere. Draw arcs passing through the extremities of the perpendicular 

line and the two points of division which are on each side of it. These 
arcs will be very nearly the outlines of the lune. 

The lunes may all be drawn and cut out and fitted over a solid 
sphere and pressed or beaten into shape. In practice it is difficult to 

fasten all the points of the lunes together and it is therefore better to cut 
them off and fill out the place with a circular piece. 

PROB. Q^. Draw the development of the surface of a 
sphere approximately by dividing it into zones. (Take sphere i}4 ins. 

in diam. and divide surface into zones 15 degrees wide.) 

Each zone may be treated as part of a cone, and its development 
drawn as in Prob. 55. The radius, with which the development of a 

zone is drawn, is an element of the complete cone, which would be tan- 
gent to the sphere at the centre of that zone. 



Proh.oZ 




Intersecting Cylinders 



Proh.58. 




PrQb.59. 



Prob.GO. 




FolyheciroTly 




MECHANICAL DRAWING. 13 

Draw a circle equal to the circumference of the sphere, and draw a 
vertical line through its centre. From one of the points where this 

line intersects the circle, divide one-fourth of the circle into 6 equal parts. 
Draw tangents to the circle from these points of division to the vertical 
line. These tangents form angles with the vertical line of 15, 30, 45, 

60 and 75 degrees respectively. The width of each zone is equal to the 

distance between the points of division. The development of the zone 

at the centre is straight, and is equal in length to the circumference of 
the sphere. The outside arcs of the developments of the two zones next 

to it are drawn with a radius equal to the Jongest tangent added to half 
the width of the zone. Their length is made equal to the circumfer- 

ence of the sphere. The inside arcs and ends of the developments are 

determined as in Prob. 55. The developments of the next two zones 

are semicircles and their outside arcs are drawn with a radius equal to 
the next longest tangent added to half the width of the zone. The out- 
side arcs of the next two are equal in length to the inside arcs of the 
semicircular ones. The outside arcs of the next smaller are equal in 

length to the inside arcs of the greater ones next to them. The open- 

ings in the smallest zones are drawn to their centres. 

If the zones are all drawn touching each other, as in the figure, the 
distance between the centres of the smallest zones is equal to one-half the 
circumference of the sphere. 

If the draftsman has a measuring scale by which he can measure dec- 
imally, and a protractor by which he can lay off angles; he can make the 
width of each zone equal to .262 times the radius of the sphere and the 
tangent lines equal respectively to 3.732, 1.732, i.ooo, .577 and .268 times 
the radius. The middle development is straight and equal in length to 
6.283 times the radius. The two developments next to it contain 93^ 

degrees; the next two 180 degrees (semicircles). The next two have 
105 degrees cut out; the next two 48^, and the two smallest 12^^ cut 
out. 






■*>^^B. 2 LIBRARY OF CONGRESS 

iillllllliillllllili*! 

IMO 019 973 643 7 






'^^'%^'^ 



'y*- 

^ % 



^^'■5t ■ .% 






*. J-»'^. *'- 



3^ 



*MJ5b^ 



->* 








e-S^ 


Z-'*.^ 

? 






M5l-r^.X' 



